Math 331 - Transformation Geometry: Syllabus and Homework

Syllabus

Textbook for most assignments: George Martin, Transformation Geometry. An Introduction to Symmetry

All problems are worth 10 points unless otherwise specified.

• Due on 2/7 Let f:R^3->R^2 and g:R^2->R^3 be given by f(e1)=e1+e2, f(e2)=2e1-e2, f(e3)=e2; g(e1)=e1-e2+e3, g(e2)=e1+3e2-2e3. Compute (fog)(3e1+2e2).
• Due on 2/14 Give an example of: 1) a linear isometry different from the identity; 2) an isometry which is not a linear map; 3) a linear map which is not an isometry; 4) an affine map which is neither an isometry nor a linear map (you might combine the previous two examples). It is easiest to think of maps on the plane R^2 to R^2 of the form f(x)=ax+b, where a is a real scalar. Justify your choices.
• Due on 2/21
  • Prove that translations and halfturns map any line {x+tv : t in R} to a parallel line (same direction v).
  • page 21/3.10 (consider halfturns in R^n, and follow the hints in class, for one of the 4 possible solutions).
• Test on 2/24 There will be 5 problems, and you need to choose 4. The extra problem counts as bonus.
  • (1) Reproducing one proof, out of the following (in the Steinberger lecture notes): Lemma 2.4.3 and Corollary 2.4.4, Proposition 2.5.1, Theorem 4.1.5.
  • (2) Affine transformations (very similar to the problem solved in the revision class).
  • (3,4) Translations, halfurns, and their compositions (practice problems on p.20-21 in the textbook).
  • (5) Calculating images under reflections.
• Due on 3/13 page 40/5.1 (read the text at the bottom of page 35 and top of 36); page 50/6.3.